ieee transactions on signal processing, vol. 00, no. 00,...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 00, NO. 00, XXX 2015 1

Linear Beamformer Design for Interference

Alignment via Rank Minimization

Gokul Sridharan, Member, IEEE, and Wei Yu, Fellow, IEEE

Abstract

This paper proposes a new framework for the design of transmit and receive beamformers for interference

alignment (IA) without symbol extensions in multi-antenna cellular networks. We consider IA in a network consisting

of G cells and K users/cell, having N antennas at each base station (BS) and M antennas at each user. The proposed

framework is developed by recasting the conditions for IA as two sets of rank constraints, one on the rank of

interference matrices, and the other on the transmit beamformers in the uplink. The interference matrix consists

of all the interfering vectors received at a BS from the out-of-cell users in the uplink. Using these conditions and

the crucial observation that the rank of interference matrices under alignment can be determined beforehand, this

paper develops two sets of algorithms for IA. The first part of this paper develops rank minimization algorithms

for IA by iteratively minimizing a weighted matrix norm of the interference matrix. Different choices of matrix

norms lead to reweighted nuclear norm minimization (RNNM) or reweighted Frobenius norm minimization (RFNM)

algorithms with significantly different per-iteration complexities. Using prior knowledge of the required rank of

interference matrices, we propose a novel weight update rule that navigates the algorithm towards aligned beamformers.

Alternately, the second part of this paper utilizes the prior rank knowledge in a different way and devises an alternating

minimization (AM) algorithm where the rank-deficient interference matrices are expressed as a product of two lower-

dimensional matrices that are then alternately optimized. Simulation results indicate that RNNM, which has a per-

iteration complexity of a semidefinite program, is effective in designing aligned beamformers for proper-feasible

systems with or without redundant antennas, while RFNM and AM, which have a per-iteration complexity of a

quadratic program, are better suited for systems with redundant antennas. All three algorithms are shown to outperform

previous state-of-the-art algorithms.

Index Terms

MIMO Beamforming, Interference Alignment, Rank Minimization, Reweighting, Alternating Minimization.

Copyright (c) 2015 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must

be obtained from the IEEE by sending a request to [email protected].

Manuscript received on July 21, 2014, revised on January 13, 2015, accepted on June 21, 2015. Date of current version July 3, 2015.

The authors are with the The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto,

Ontario, Canada, M5S3G4, e-mail: ([email protected], [email protected]).

This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Ontario Centre of Excellence, and

BLiNQ Networks Inc. The material in this paper was presented in parts at the IEEE International Conference on Acoustics, Speech and Signal

Processing (ICASSP), Vancouver, Canada, May, 2013, and the IEEE Workshop on Signal Processing Advances in Wireless Communications

(SPAWC), Toronto, Canada, June, 2014.

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I. INTRODUCTION

Linear beamforming techniques for interference mitigation are of significant interest in MIMO cellular networks

where increasing base-station (BS) density along with backhaul enhancements have necessitated and enabled coor-

dinated interference management. In this context interference alignment has emerged as a key concept in addressing

interference in such networks. In contrast to asymptotic interference alignment [1] that requires decomposition of

multi-antenna nodes and infinite symbol extensions, linear beamforming based alignment schemes are simpler to

implement and therefore more relevant in practice. In this work we develop algorithms to design aligned transmit

and receive beamformers to achieve a requisite number of degrees of freedom (DoF) in a G-cell, K-users/cell

network with N antennas at each base station (BS) and M antennas at each user—a (G,K,N × M) network,

assuming channels to be generic with no symbol extensions.

Our interest in designing algorithms for interference alignment is twofold. First, in cooperative cellular networks

that operate in the interference-limited regime, these beamformers identify regions in the optimization landscape

where interference is significantly mitigated. Second, while techniques from algebraic geometry can be used to

establish feasibility of interference alignment [2]–[5], techniques for the actual design of aligned beamformers still

rely on numerical approaches. Despite the availability of non-iterative approaches based on notions such as the

subspace alignment chains for the 3-user interference channel [6] or the unstructured approach [7] for any G-cell,

K-users/cell network but under certain restrictions on N , M , and the number of DoF required per user, iterative

algorithms [8]–[16] continue to play an important role for general network parameters without any restrictions.

In this paper we develop two novel sets of iterative algorithms to design aligned beamformers. A crucial insight of

this paper is that exploiting prior knowledge of the rank of interference matrices significantly benefits the numerical

convergence of the algorithms. The interference matrix consists of all the interfering vectors received in the uplink

at a BS from the out-of-cell users. In the first set of algorithms, we use the prior rank knowledge to develop

an effective algorithm to solve a rank minimization formulation. In an alternate approach, we restrict the rank of

interference matrices by expressing them in a product form and develop a computationally efficient algorithm to

find aligned beamformers.

A. Literature Survey

Several iterative algorithms are available to design beamformers for interference alignment [8]–[18]. In [8], an

iterative algorithm called iterative leakage minimization (ILM) is proposed for the MIMO interference channel. It is

based on minimizing the sum of interference powers at all the receivers. This algorithm is extended to MIMO cellular

networks in [10]. While algorithms of [8] and [10] are known to converge to a locally optimal solution for minimizing

interference power, empirically it is often the case that they need several hundred iterations to converge. However,

their per-iteration complexity is low as they only require computing a small number of eigenvalue decompositions

per iteration. More recently, [13], [14] also propose algorithms that minimize the total interference power across

all the receivers. In [13], the total interference power is minimized while imposing a reduced row echelon form on

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the transmit and receive beamformers to ensure that they are full rank. In [14] a message passing framework to

minimize total interference power is established and ILM is interpreted to be a special case of this framework.

In [11], a rank-constrained rank minimization (RCRM) framework for finding linear beamformers for interference

alignment is proposed. Recognizing that the algorithms proposed in [8], [10] are equivalent to minimizing the

Frobenius norm (matrix analog of the ℓ2-norm for vectors) of a projection of the interference matrix, the authors

instead propose to minimize its nuclear norm. The projected interference matrix is the projection of the uplink

received interference vectors at a BS on the useful signal space as determined by the receive beamforming vectors

at each BS. Nuclear norm minimization (the matrix analog of the ℓ1-norm minimization for vectors) is known to

induce sparsity and is well suited for generating aligned solutions. This framework is also capable of designing

aligned beamformers when time or frequency extensions are allowed. Using the same framework, [15] and [16]

further propose a reweighted version of nuclear norm minimization for finding aligned beamformers. Due to the

nuclear norm approximation, the algorithms of [11], [15], [16] involve solving a series of semidefinite programs

(SDPs). These algorithms are computationally more intensive, on a per-iteration basis, than the algorithms of [8],

[10], but they require fewer iterations. However, in spite of using approximations which are sparsity inducing, these

algorithms still lack explicit mechanisms to generate the desired level of sparsity for interference alignment and

often fall short of generating the requisite number of interference free dimensions in practice—a shortcoming this

paper seeks to address.

A key difference between [11], [15], [16] and this paper is that this paper works directly with the interference

matrix rather than its projection using the receive beamformers. Note that the rank of the projected interference

matrix when using aligned beamformers is zero. Clearly, this places an extreme sparsity requirement on the above

approaches. As seen later in this paper, working directly with the interference matrices allows us to employ novel

mechanisms to control the desired level of sparsity of the interference matrices.

Different from the above approaches, [17] and [18] develop algorithms to create sufficient number of interference-

free dimensions for signal-vectors by minimizing a subset of eigenvalues of the interference matrix subject to a

unitary constraint on the uplink transmit beamformers. The unitary constraint requires the optimization to be carried

out on the complex Steifel manifold.

B. Proposed Framework

In this work, we focus on linear interference alignment for generic channels without symbol extensions and

reformulate the conditions for interference alignment as two sets of rank constraints, the first imposed on the

interference matrix of each BS and the second on the transmit beamforming matrices. This framework is inspired

by counting arguments that partition the total number of dimensions at each BS as either being occupied by signal

or interference. The algorithms in [17], [18], developed originally for the MIMO interference channel, represent

one particular approach to designing algorithms for interference alignment using such a framework.

Using this framework we develop three distinct and novel algorithms that aim to design uplink transmit beam-

formers for interference alignment. The first two algorithms, namely, the reweighted Frobenius norm minimization

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(RFNM) and reweighted nuclear norm minimization (RNNM) are developed by recasting the rank constraints as a

rank minimization problem subject to linear constraints, while the third algorithm, called alternating minimization

(AM), is developed by expressing rank-deficient matrices that satisfy the rank constraints as a product of two lower-

dimensional matrices, then solving a convex quadratic program. In each of these cases, this paper makes a key

observation that exploiting prior knowledge of the rank of interference matrices is crucial and that such information

can be inferred based on the network parameters and the required DoF for all users, as long as these DoF/user are

known to be achievable.

Establishing the achievability of a certain DoF/user in a given (G,K,N×M) network using linear beamforming

without symbol extensions is actually a nontrivial problem. Efforts to establish achievability have lead to techniques

based on algebraic geometry that provide an answer to feasibility without explicitly designing the beamformers [2]–

[5]. The primary focus of this paper is to design aligned beamformers after the question of feasibility has been

resolved. Thus, while we significantly rely upon previous works that aim to ensure feasibility, we also go beyond

these papers and provide effective mechanisms to design the aligned beamformers.

1) Reweighted Matrix Norm Minimization: The rank minimization formulation aims to minimize the total number

of dimensions occupied by interference at each BS by minimizing the rank of the interference matrix. Minimizing

the dimensions occupied by interference opens up more dimensions for signal vectors. While the formulation

in this paper is similar in spirit to [11], [17], [18], there are also significant differences. First, unlike [11], we

only optimize over uplink transmit beamformers and eliminate the need to alternately optimize over transmit and

receive beamformers. Second, instead of imposing rank constraints or unitary constraints on the uplink transmit

beamformers, we impose a simple set of affine constraints that are straightforward to satisfy. These changes allow

us to pose the problem of designing beamformers for interference alignment as a rank minimization problem subject

to affine constraints only.

Rank minimization subject to affine constraints is well studied in the context of compressive sensing and there

exist several effective algorithms to solve such problems [19]–[25]. Drawing inspiration from these algorithms, we

solve the rank minimization problem for interference alignment by approximating rank using a series of weighted

Frobenius or nuclear norms. The iterative reweighting of the matrix norms provides improved approximation to the

rank of the interference matrix after each iteration.

A crucial observation made in this paper is that utilizing prior rank knowledge of the interference matrix can

significantly improve the convergence of the algorithms. Towards this end, we propose a novel reweighting rule that

couples the weights in the weighting matrices by taking advantage of the prior knowledge of the expected rank of

interference matrices. This coupling of weights acts as a control mechanism to indicate to the algorithm the desired

level of sparsity, i.e., the desired ranks of the interference matrices.

The choice between reweighting the Frobenius and the nuclear norm results in two computationally different

algorithms: RFNM, which minimizes a Frobenius norm, requires solving an unconstrained convex quadratic program

at every iteration, and RNNM, which minimizes a nuclear norm, requires solving a semidefinite program. Although

the per-iteration complexity of RNNM is significantly higher than that of RFNM, RNNM requires much fewer

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iterations to converge. This raises an interesting trade-off between complexity-per-iteration and the total number of

iterations required.

2) Alternating Minimization: Recognizing that the rank contraints on the interference matrices is the crucial

bottleneck in designing beamformers for interference alignment, this paper further devises an alternative to rank

minimization algorithms by explicitly imposing the rank constraint on the interference matrices. Since the desired

ranks of the interference matrices are known a priori, we can express these rank-deficient matrices as a product

of two lower-dimensional matrices. Once expressed in this manner, we can alternately optimize one of the two

component matrices while holding the other constant. Such an approach, called alternating minimization (AM), has

been previously proposed in the context of robust rank minimization in the presence of noisy observations [26],

[27]. The optimization procedure can be interpreted as iterative minimization of the difference between the actual

interference matrix as determined by the set of uplink transmit beamformers and a nominal interference matrix that

identifies the subspace reserved for interference.

C. Summary of Contributions

The main contributions in this paper include a new framework for interference alignment using which we develop:

(a) Two new algorithms for interference alignment based on rank minimization subject to affine constraints; (b) An

algorithm that uses alternating minimization to design aligned transmit beamformers in the uplink that satisfies the

necessary rank constraints by expressing each interference matrix as a product of two lower-dimensional matrices. In

both cases, exploiting the prior knowledge of the rank plays an important role in the effectiveness of the algorithms.

Key insight on system design from simulation results includes the following: (a) having redundant antennas in the

network significantly reduces the complexity of computing aligned beamformers—algorithms with low per-iteration

complexity such as RFNM, AM and ILM are very effective in this case; (b) networks with no redundant antennas are

the more challenging ones for designing aligned beamformers—the only effective algorithm for designing aligned

beamformers for such networks is RNNM, where per-iteration complexity is that of an SDP.

D. Paper Organization

This paper is organized as follows. In Sections II and III we introduce the system model and present a reformu-

lation of the conditions for interference alignment. In Section IV, using this reformulation, we pose the problem of

designing aligned beamformers as a rank minimization problem and develop the RFNM and RNNM algorithms. In

Section V we develop the AM algorithm. Section VI compares the proposed algorithms with existing algorithms

and Section VII presents the simulation results.

E. Notation

Column vectors are denoted by bold lower-case letters and matrices by bold upper-case letters. The conjugate

transpose and Euclidean norm of vector v are denoted as vH and ‖v‖, respectively. The column span of the columns

of a matrix X is denoted as span(X). The Frobenius norm and nuclear norm are denoted as ‖ · ‖F and ‖ · ‖∗

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respectively. We use the simplified notation {Xij} to represent a set of matrices Xij where the indices i and j are

within a range which is typically clear from the context. Similarly, we use the notation {Xij}Lj=1 when the index

i is held constant and j is varied from 1 to L. We use the notation X(1 : i, 1 : j) to refer to the submatrix of X

consisting of elements from the first i rows and first j columns of X. The set of positive integers and the set of

complex numbers are represented as Z+ and C, respectively.

II. SYSTEM MODEL

Consider the uplink of a cellular network consisting of G interfering cells with K users per cell. Each user is

assumed to have M antennas and each BS is assumed to have N antennas. Let the channel from the jth user in the

ith cell to the gth base station (BS) be denoted as the N×M matrix H(ij,g). We assume all channels to be generic.

Equivalently, the channels can be assumed to be randomly drawn according to a cumulative distribution function in

CG2KMN that is absolutely continuous and is such that all the conditional cumulative distribution functions of this

distribution are also absolutely continuous. All channels are assumed to be known perfectly at a central location.

The M × 1 signal vector transmitted by the kth user in the gth cell is formed using a M × dgk linear transmit

beamforming matrix Vgk and received using a N × dgk receive beamforming matrix Ugk , where dgk represents

the number of transmitted data streams of the kth user in the gth cell. The received signal after being processed

by the receive beamforming matrix Ugk at the gth BS can be written as

UHgkyg =

G∑

i=1

K∑

j=1

UHgkH(ij,g)Vijsij +UH

gkng, (1)

where sij is the dij×1 symbol vector transmitted by the jth user in ith cell and ng is the N×1 vector representing

circular symmetric additive white Gaussian noise ∼ N (0, I). In this paper, we restrict our focus to the symmetric

case where dgk = d ∀g, k. However, the same framework can also be extended to the asymmetric cases. The

downlink received signal is defined similarly.

Although the development of the proposed algorithms is presented using the uplink signal model, the aligned

beamformers that are eventually designed can be used in either uplink or downlink because of the duality between

uplink and downlink for interference alignment [8].

We denote the space occupied by interference at the gth BS as the column span of a N× (G−1)Kd interference

matrix Rg formed using the column vectors from the set {H(ij,g)vijl : i ∈ {1, 2, . . . , G}, j ∈ {1, 2, . . . ,K}, l ∈

{1, 2, . . . , d}, i 6= g}, where we use the notation vijl to denote the lth beamformer associated with the jth user in

the ith cell. For example, for the (3, 2, 4× 3) network with 1 DoF/user, R1 is a 4× 4 matrix given by

R1 =[H(21,1)v21 H(22,1)v22 H(31,1)v31 H(32,1)v32

]. (2)

To enable the ability to measure and collect channel state information at a centralized location, this paper assumes

the presence of backhaul links connecting all the BSs. These backhaul links allow the aligned beamformers to be

communicated to each of these BSs. The BSs can use these beamformers in the downlink, over the air, so that the

users can then locally compute their receive beamformers to null all the interference, thus achieving alignment.

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III. PROBLEM FORMULATION

The conditions for linear interference alignment when symbol extensions over time or frequency are not allowed

can be stated as follows [28]:

UHgkH(ij,g)Vij = 0 ∀(i, j) 6= (g, k) (3)

rank(UHgkH(gk,g)Vgk) = d ∀g, k. (4)

Interference alignment is said to be feasible if there exist transmit and receive beamformers (Vgk and Ugk) that

satisfy the above two conditions. Counting the number of equations and variables involved in these conditions gives

a preliminary check on the feasibility of satisfying (3) and (4). When the number of variables exceed the number

of equations, the system is said to be proper. A (G,K,N ×M) network where d DoF/user are required is proper if

(M+N) ≥ (GK+1)d [10], [28]. While not all proper systems are feasible [6], improper systems have been shown

to be almost surely infeasible [2]. This paper only considers proper systems whose feasibility has been established

either through the techniques in [2]–[5], or by other means.

The conditions for interference alignment can be alternately stated as

rank(Rg) ≤ N −Kd ∀g, (5)

rank(Vgk) = d ∀g, k. (6)

This reformulation follows from the fact that when interference spans no more than N −Kd dimensions, generic

channels ensure that the intersection between useful signal subspace (span({H(gk,g)Vgk}Kk=1)) and interference

subspace (span(Rg)) is almost surely zero dimensional. This allows us to replace (3) with a rank constraint on the

interference subspace (5). In addition, since generic channel matrices are almost always full rank, condition (4)

is satisfied as long as rank(Vgk) = d. This is because direct channels do not play a role in (5) and (6), and if

rank(Vgk) = d then rank(H(gk,g)Vgk) = d almost surely and further, rank([H(g1,g)Vg1, . . . ,H(gK,g)VgK ]) =

Kd as well. This allows intra-cell interference to be nulled completely without losing signal dimensions, thus

ensuring the existence of a valid set of receive beamformers at each BS. We collectively refer to the set of {Vgk}

as V.

Since we need to design transmit beamformers that satisfy conditions (5) and (6), it is natural to pose the problem

of finding these beamformers as the following feasibility problem:

minimize 1

subject to rank(Rg) ≤ N −Kd ∀g,

rank(Vgk) = d ∀g, k,

Ag(V) = Rg ∀g.

(7)

where Ag(V) = (Rg) captures the linear relationship between V and Rg as shown in (2). While the rank constraint

on Vgk is easily handled by restricting it to be in column-reduced echelon form, i.e., Vgk(1 : d, 1 : d) = I ∀g, k,

handling the rank constraint on Rg is not straightforward. Optimization problems involving rank constraints are

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typically difficult to solve, requiring algorithms to traverse the manifold of low-rank matrices, or requiring the

exploitation of the problem structure to recast the problem into a more amenable form [29]–[31]. For the problem

under consideration, we exploit the prior knowledge of the rank of Rg and re-frame the rank constraint into

numerically more amenable formulations.

Towards this end, we propose two contrasting formulations. We first note that it is possible to replace the

rank constraints on Vgk with equivalent affine constraints without any loss of generality; this leaves us with

only one set of rank constraints to satisfy. In the first formulation we transform the feasibility problem to a rank

minimization problem by moving the rank constraints on Rg to the objective function. Such a transformation leads

to an optimization problem of the form

minV

maxg∈{1,2,...,G}

rank(Rg)

subject to Vgk(1 : d, 1 : d) = I ∀g, k (8)

Ag(V) = Rg ∀g.

This is a minimax optimization problem where we minimize the maximum rank of the interference matrices

R1,R2, . . . ,RG. Since the system is assumed to be proper and feasible, the global optimum of this optimization

problem is no more than N − Kd. Further, any set of beamformers that achieves the objective rank of N −Kd

constitutes a set of aligned solutions. As shown in the next section, such prior knowledge of rank allows us to

devise effective algorithms to find aligned beamformers. This prior knowledge is rarely available in generic rank

minimization problems but is available in the context of interference alignment based on the network parameters

and the required DoF for all the users. This prior knowledge proves vital in reducing just the right number of

singular values of the interference matrices to zero.

In the second formulation we relax the affine constraint governing V and Rg and move this constraint to the

objective function by imposing a quadratic penalty on their difference (Ag(V) −Rg). This allows us to treat Rg

as a free variable which is independent of V and lets us handle the rank constraint on Rg by expressing it as a

product of two matrices Pg and Qg where Pg is a N × (N −Kd) matrix and Qg is a (N −Kd)× (G− 1)Kd

matrix, so that the rank of Rg is no more than N −Kd. This reformulation leads to an optimization problem of

the form

minV,{Pi},{Qi}

maxg∈{1,2,...,G}

‖Ag(V) −Rg‖2F

subject to Rg = PgQg ∀g, (9)

Vgk(1 : d, 1 : d) = I ∀g, k.

Since all matrices permit a singular value decomposition (SVD), there is no loss of generality in expressing Rg as

the product PgQg . Since the network is assumed to be proper and feasible, the global optimum of this optimization

problem occurs when the objective is reduced to zero. The set of beamformers obtained at the global optimum

satisfy the conditions for interference alignment.

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Note that the two formulations differ in the constraint set over which the matrix Rg is optimized. While the

second formulation restricts the optimization to a set of rank deficient matrices, no similar rank condition is imposed

in the first. Since rank deficient matrices constitute a measure-zero set over the space of all matrices of a given

dimension, restricting the search space to a set of rank deficient matrices significantly prunes the search space.

However, this gain comes at the cost having to define Rg as a product of two matrices, thereby losing linearity.

Due to the non-smooth, non-convex nature of the rank function (8), the non-linearity in (9), and the difficulty in

dealing with minimax formulation in both cases, further approximations to these formulations are necessary before

developing algorithms to find aligned beamformers. We treat these two formulations in further detail in the next

two sections.

IV. INTERFERENCE ALIGNMENT AS RANK MINIMIZATION

This section focuses on the first formulation where interference alignment is cast as a rank minimization problem.

We first propose to replace the minimax formulation (8) by a min-sum formulation given by

minimizeV

G∑

g=1

rank(Rg)

subject to Vgk(1 : d, 1 : d) = I ∀g, k

Ag(V) = Rg ∀g.

(10)

Although techniques such as the subgradient algorithm can be used to solve the minimax formulation directly, such

techniques are known to converge slowly. Replacing the minimax optimization problem with a min-sum optimization

problem significantly alters the problem landscape. However, this paper shows that by taking advantage of the

prior knowledge of the rank of optimal Rg, this min-sum formulation can be effectively used to design aligned

beamformers. This addresses a key difficulty in dealing with the original minimax formulation (8). A similar

approach is adopted for solving (9) later in the paper.

Next, in order to apply standard optimization techniques for rank minimization, rank is approximated using a

surrogate function. Well-known surrogate functions for rank include the nuclear norm (convex envelope of rank), the

Schatten-p function [25], the log(det(·)) function and the −tr(inv(·)) function [23], [24]. The choice of surrogate

function determines the per-iteration complexity of the resulting algorithms. Solving (10) using the log(det(·))

approximation leads to a sequence of SDPs each minimizing a weighted nuclear norm [19], while the Schatten-p

function requires solving a series of unconstrained convex quadratic programs each minimizing a weighted Frobenius

norm. Either approximation leads to solving a series of convex optimization problems of the form

minimizeV

G∑

g=1

f (s)g (Rg)


Ag(V) = Rg ∀g.

(11)

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where f(s)g (Rg) is a convex function of Rg that is used to approximate its rank in the sth iteration. The crucial

element of such an approach is the iterative update of f(s)g (·). While the iterative updates suggested in [19], [25]

are effective for rank minimization problems, they do not always minimize the rank of Rg to the required extent

for interference alignment in the min-sum formulation. The main idea of this paper is to develop a new update rule

that ensures that the rank of Rg is minimized to the desired extent while factoring in the minimax nature of the

original formulation. This new approach is discussed for weighted Frobenius norm and for weighted nuclear norm

minimization separately below.

A. Rank Minimization Using Frobenius Norm

1) RFNM Algorithm: We begin by discussing the standard approach for matrix rank minimization via reweighted

Frobenius norm minimization. Solving affine-constrained rank minimization by iteratively solving a series of

quadratic programs that minimize a weighted Frobenius norm is first discussed in [32], where the rank of a matrix

X is approximated using the Schatten-p function given by tr(XHX + γI)p/2 for 0 < p ≤ 1. Noting that the

derivative of the Schatten-p function is given by pX(XHX + γI)p/2−1, it is shown that the KKT conditions of

the resulting affine-constrained optimization problem can be satisfied by iteratively solving a set of weighted-least-

squares problems. Mathematically, the affine constrained rank minimization problem:

minimize rank(X)

subject to A(X) = b,(12)

is solved by iteratively solving the following quadratic optimization problem:

minimize ‖X(W(s))1/2‖2F

subject to A(X) = b,(13)

where the weights W(s) are positive semidefinite matrices and are updated using the rule W(s+1) = ((X(s))H(X(s))+

γ(s+1)I)p2−1 where the optimal X obtained after the sth iteration is denoted as X(s) and γ(s+1) is the regularization

parameter used in updating the (s + 1)th weight. Further, the same iterative procedure can also be applied when

p = 0, where the weight update rule is justified by showing that it solves a fixed point equation emerging from the

KKT conditions that result when rank of X is approximated as log(det(XHX+ γI)).

When weights are updated using the update rule given above, the weighting matrices can be interpreted to weight

the singular values of the matrix (X(s))HX(s). To see this, let the singular value decomposition of X(s) be given by

P(s)Σ(s)(Q(s))H , then W(s+1) = Q(s)((Σ(s))p−2 + γ(s+1)I)−1(Q(s))H . Thus when p = 0, the weighting matrix

W(s+1) imposes a penalty that is inversely proportional to the square of the magnitude of each non-zero singular

value of X(s). Since small, non-zero singular values are heavily penalized, the iterative procedure is incentivized

to reduce them to zero, thus reducing the rank of X.

To solve (10), we adapt the RFNM approach outlined above, with p = 0, by proposing a novel reweighting

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Algorithm 1 Reweighted Frobenius Norm Minimization

1: Set γmin = 10−8, s = 1; Initialize W(1)g = I ∀g.

2: while s ≤ itermax do

3: Solve (14) using weights W(s)g and denote the optimal interference matrices as Ropt

g .

4: Compute the reduced SVD of Roptg , and denote it as P

(s)g Σ(s)

g (Q(s)g )H .

5: Set σ2min = ming,r σ

2(g,r), where σ(g,r) are the singular values of R

(s)g arranged in descending order.

6: Set D(s)g = diag([σ2

(g,1), . . . , σ2(g,N−Kd), σ

2min, . . . , σ

2min

︸︷︷︸

z times

]).

7: Set γ(s+1) = max(σ2min, γmin).

8: Update W(s+1)g = (Q

(s)g D

(s)g (Q

(s)g )H + γ(s+1)I)−1.

9: Set s = s+ 1.

10: end while

strategy. This approach requires iteratively solving an optimization problem of the form

minimizeV

G∑

g=1

‖Rg(W(s)g )1/2‖2F


Ag(V) = Rg ∀g.

(14)

where Rg is weighted using W(s)g in the sth iteration. Note that we implicitly assume that Rg has more rows

than columns, if not, we simply replace Rg with RHg in the above formulation and all the subsequent steps. This

ensures that the number of singular values of W(s)g and Rg are the same. It turns out that a direct application of

the weighting procedure outlined above to (14), while inducing a low-rank Rg, almost never generates the total

requisite number of interference-free dimensions. For example, the formulation in (14) may lead to scenarios where

we have more than the necessary number of interference free dimensions at one BS with insufficient interference-

free dimensions at other BSs. These issues are effectively addressed by the new reweighting technique detailed

below that exploits the prior knowledge of the required rank of Rg.

2) Proposed Reweighting Technique: Since we are looking for transmit beamformers that ensure rank(Rg) ≤

N −Kd ∀g, we require z = min(Kd, (GKd−N)) singular values of Rg to be zero. To avoid local minima where

rank of Rg is not sufficiently minimized, we couple the penalties (weights) associated with each of the z smallest

singular values of R1,R2, . . . ,RG. Let {σ(g,r) : r = 1, 2, . . . ,min ((G− 1)Kd,N)} be the set of singular values

of Rg obtained after the sth iteration, ordered in the descending order i.e., σ(g,r) ≥ σ(g,r+1). Further, let

σ2min = min

g,rσ2(g,r), (15)

July 20, 2015 DRAFT


and define the diagonal matrix D(s)g as

D(s)g = diag([σ2

(g,1), σ2(g,2), . . . ,σ

2(g,(N−Kd)),

σ2min, σ

2min, . . . , σ

2min

︸︷︷︸

z times

]). (16)

We set the weights for the (s+ 1)th iteration to be

W(s+1)g = (Q(s)

g D(s)g (Q(s)

g )H + γ(s+1)I)−1, (17)

where Q(s)g is the matrix of right singular vectors obtained from an SVD of the optimal Rg obtained after the sth

iteration. Such an update equally penalizes each of the z smallest singular values of Rg ∀g, thereby encouraging

the algorithm to seek aligned beamformers where all z smallest singular values of each Rg can be simultaneously

set to zero. The proposed iterative procedure is summarized as Algorithm 1.

The parameter γ(s+1) acts as a regularization constant that makes sure the weighting matrices are positive definite.

It limits the penalty imposed on small non-zero singular values and is typically reduced with each iteration to prevent

the algorithm from prematurely converging to local minima.

Once the transmit beamformers in the uplink are designed, the receive beamformers at the gth BS for recovering

the data streams of the kth user can be chosen to be the left-singular vectors corresponding to the d smallest singular

values of the matrix Rg augmented with the interfering vectors from other users in the same cell. That is, suppose

PΣQH represents the SVD of the matrix

[Rg H(g1,g)Vg1 . . .H(g(k−1),g)Vg(k−1) H(g(k+1),g)Vg(k+1)

. . .H(gK,g)VgK ],

where Σ has singular values listed in the decreasing order, then we set

Ugk = P(1 : N,N − d+ 1 : N). (18)

Such a choice of uplink receive beamformers ensures that both inter-cell and intra-cell interference are minimized

for the given set of uplink transmit beamformers. When the uplink transmit beamformers are the aligned beamform-

ers, this choice of receive beamformers completely nulls out inter-cell and intra-cell interference, thus achieving

alignment.

3) Convergence and Complexity: When weights are updated according to the original update rule given in [25]

with p = 0, the value of the surrogate function can be shown to converge; but a proof of convergence of the iterates

themselves is not yet known. For the proposed weight update rule that couples the weights, proof of convergence

of the surrogate function is also not available. However, since decoupling the weights ensures convergence, we can

decouple the weights after a fixed number of iterations to let the surrogate function converge. In our simulations

we simply run the algorithm for a fixed number of iterations (itermax) with coupled weights.

To analyze the complexity of the algorithm, note that each iteration of the proposed heuristic requires solving

the quadratic program (14) and computing SVDs of G matrices of size N × (G− 1)Kd. The two linear constraints

July 20, 2015 DRAFT


in (14) are easy to eliminate as Rg is just an auxiliary variable and the conditions on Vgk are easy to satisfy.

Thus, (14) is an unconstrained quadratic program in ν = GK(M − d)d variables and requires solving a linear

system in the same number of variables which can be accomplished in O(ν3) time. Each SVD can be computed

in O(max(N, (G− 1)Kd)3) time.

B. Rank Minimization Using Nuclear Norm

1) RNNM Algorithm: We now present an analogous reweighting strategy for solving the interference align-

ment problem by rank minimization where the rank is approximated by nuclear norm. Rank minimization is the

matrix analog of ℓ0-norm minimization for vectors, and techniques developed for ℓ0-norm minimization can be

extended to rank minimization. In the compressive sensing literature, ℓ0-norm minimization is often approximated

by reweighted ℓ1-norm minimization [20]. The matrix counterpart of reweighted ℓ1-norm minimization proposed

in [20] is reweighted nuclear norm minimization [19]. This procedure reformulates the rank minimization in (10)

as

minimizeV

G∑

g=1

‖WL,gRgWR,g‖∗


Ag(V) = (Rg) ∀g.

(19)

where WL,g and WR,g are two positive definite matrices that are interpreted to reweight the nuclear norm of Rg.

The iterative procedure involves solving an instance of (19) for fixed WL,g and WR,g, then updating the weights

for the next iteration. Every iteration requires solving a semidefinite program.

The choice of the weight update rule affects the overall performance and needs to be chosen carefully. In [19],

the authors derive a weight update rule by exploiting an equivalence relation between the rank of Rg and a positive

semidefinite matrix Z of the form

A Rg

RHg B

, and approximating the rank of Z using the log(det(·)) surrogate

function. The weight update rule is interpreted to minimize the concave surrogate function through a majorization-

minimization procedure [19], [20], [23]. The resulting weights can be interpreted to reweight the singular values

of the matrix Z.

2) Proposed Reweighting Technique: While we adopt the general framework of reweighting as proposed in [19],

we update weights using an update rule similar to that proposed in the previous section. Let z = min(Kd, (GKd−

N)) and let y = min(N, (G − 1)Kd). Note that we require z of the y singular values of Rg to be zero. Let

P(s)g Σ(s)

g (Q(s)g )H be the full singular value decomposition of the optimum Rg obtained after the sth iteration

and let σmin = ming,k σ(g,k), where σ(g,k) are the singular values of Σ(s)g . We once again couple the penalties

associated with those singular values that need to be set to zero and update the weights for the (s+ 1)th iteration

as follows

W(s+1)L,g =P(s)

g (D(s+1)L,g + γI)−

12 (P(s)

g )H (20)

W(s+1)R,g =Q(s)

g (D(s+1)R,g + γI)−

12 (Q(s)

g )H (21)

July 20, 2015 DRAFT


Algorithm 2 Reweighted Nuclear Norm Minimization

1: Set γ = 10−6, s = 1; Initialize W(1)L,g = I, W

(1)R,g = I ∀g.


3: Solve (19) using weights W(s)L,g and W

(s)R,g; Denote the optimal interference matrices as R

(s)g .

4: Compute the full SVD of R(s)g , and denote it as P

(s)g Σ(s)

g (Q(s)g )H .

5: Set σmin = ming,r σ(g,r), where σ(g,r) are the singular values of R(s)g arranged in descending order.

6: Set D(s)L,g and D

(s)R,g as given in (22) and (23).

7: Compute W(s+1)L,g and W

(s+1)R,g using (20) and (21).

8: Set s = s+ 1.

9: end while

where D(s+1)L,g and D

(s+1)R,g are defined as

D(s+1)L,g =diag

([σ(g,1), σ(g,2), . . . , σ(g,y−z),

σmin, σmin, . . . , σmin︸︷︷︸

N−y+z times

])

(22)

D(s+1)R,g =diag

([σ(g,1), σg,2, . . . , σ(g,y−z),

σmin, σmin, . . . , σmin︸︷︷︸

(G−1)Kd−y+z times

])

(23)

where the singular values are assumed to be ordered in the descending order. Again, the idea is to equally penalize

the singular values that are to be reduced to zero so that Rg is of the requisite rank. The regularization parameter

γ is set to be constant, e.g., 10−6. The proposed iterative procedure is summarized as Algorithm 2.

3) Convergence and Complexity: A key difference between RFNM and RNNM is that every iteration of RNNM

requires solving an SDP. SDPs are typically solved using the primal-dual interior point method [33], as in popular

solvers such as SDPT3 [34], [35]. The SDP in (19) requires minimizing the rank of a block diagonal matrix

consisting of G blocks each of size N × (G− 1)Kd. This problem can be posed in the standard SDP form using

Lemma 1 of [36]. Excluding auxiliary variables such as Rg, when posed in standard form there are a total of νsdp

free variables, where

νsdp = GKd(M − d) +N2/2 + ((G− 1)Kd)2/2. (24)

The per-iteration complexity of the primal-dual interior point method used in generic solvers when applied to

RNNM grows at least as fast as O(ν3sdp) [37]. Custom SDP solvers that exploit structure in the SDP can reduce

the per-iteration complexity to O(νsdp−custom) [37], where νsdp−custom is given by

νsdp−custom = N × (G− 1)Kd× (GK(M − d)d)2. (25)

July 20, 2015 DRAFT


In general, the number of iterations of the primal-dual interior point method required to solve one instance of

(19) is between 10-50 iterations and is only weakly dependent on the network parameters. After solving the SDP,

further computations are necessary to compute the G SVDs to reweight the nuclear norm. Clearly, the complexity

of solving one instance of (19) is considerably higher than that of (14). However, RNNM only requires solving

about 20-40 instances of (19) as opposed to several hundred instances of (14) in the case of RFNM. It is hence

difficult to compare the overall complexity of RFNM and RNNM.

Although a proof of convergence for RNNM using the proposed weight update rule is not yet available, simulation

results indicate that the algorithm converges within tens of iterations. In our simulations we run the algorithm for

a fixed number of iterations (itermax).

V. INTERFERENCE ALIGNMENT USING AM

A. AM Algorithm

We now turn to the second formulation (9) for the interference alignment problem. As seen in (9), relaxing the

linear constraint between the interference matrix Rg and the transmit beamformers V by imposing a quadratic

penalty on the difference allows us to treat Rg as a free variable and to express it as the product PgQg. This new

approach to satisfying rank constraints on Rg is inspired by the alternating minimization algorithm proposed in

the context of robust rank minimization [26], [27]. Such an approach proposes to iteratively optimize either Pg or

Qg while holding the other constant. With computational complexity in mind, we again simplify (9) to a min-sum

formulation as given below:

minimizeV,{Pg},{Qg}

G∑

g=1

||Ag(V) −Rg||2F

subject to Rg = PgQg ∀g,

Vgk(1 : d, 1 : d) = I ∀g, k.

(26)

Note that the global optimum is still zero and that all sets of aligned beamformers achieve the global optimum.

When either Pg or Qg is fixed, the above formulation is a convex quadratic program. We solve (26) by alternately

solving for Pg and Qg while keeping the other fixed. Intuitively, Pg and Qg jointly define the space occupied by

interference at the gth BS, and the beamformers are designed to make sure that the residual interference that spills

beyond the space identified by Pg and Qg is minimized at every iteration. The proposed algorithm is summarized

in Algorithm 3.

B. Convergence and Complexity

Since every step in alternating minimization decreases the quadratic objective, the value of the objective converges

as it is bounded below by zero (although again proof of convergence of iterates Pg, Qg and Vgk themselves is not

yet available).

To analyze the complexity of the proposed algorithm, note that the equality constraints in (27) and (28) are

trivial to satisfy. Solving (27) and (28) requires finding the minimum of an unconstrained convex quadratic function

July 20, 2015 DRAFT


Algorithm 3 Alternating Minimization

1: Initialize Qi for i ∈ {1, 2, . . . , G} with random entries drawn from a continuous distribution. Set s = 1.


3: Solve the following optimization problem using Qi obtained from Step 4 (for the first iteration use Qi

initialized to random entries):

minimizeV,{Pg}

G∑

g=1

||Ag(V)−PgQg||2F

subject to Vgk(1 : d, 1 : d) = I ∀g, k.

(27)

4: Solve the following optimization problem using Pi obtained from Step 3.

minimizeV,{Qg}

G∑

g=1

||Ag(V) −PgQg||2F


(28)

5: Set s = s+ 1.

6: end while

involving νp = GKd(M − d) +GN(N −Kd) complex scalar variables in the case of (27) and νq = GKd(M −

d) + G(G − 1)Kd(N − Kd) complex scalar variables in the case of (28). This requires solving a set of linear

equations involving νp and νq variables respectively and the worst-case complexity of solving this set of linear

equations is given by O(ν3p) and O(ν3q ) respectively. This algorithm is comparable in complexity to RFNM as it

requires solving a simple quadratic program. In addition, it does not require computing any SVDs.

VI. COMPARISON TO EXISTING ALGORITHMS

We provide a brief discussion on some of the existing algorithms for the purpose of performance comparison.

Two well-known algorithms for interference alignment are ILM [8]–[10] and RCRM [11]. ILM aims to minimize

the overall interference power across the network by solving the following optimization problem:

minimizeVgk,Ugk

G∑

g=1

‖[Ug1, . . . ,UgK ]HRg‖2F

subject to rank(Vgk) = d ∀g, k

rank([Ug1, . . . ,UgK ]) = Kd ∀g

(29)

Note that only inter-cell interference is taken into account as intra-cell interference can be nulled subsequently by

an appropriate linear transformation of the receive beamformers ({Ugk}). The algorithm alternately optimizes the

transmit and receive beamformers and involves computing G(K + 1) eigenvalue decompositions in each iteration.

Interpreting interference alignment as reducing the rank of the projected interference matrix, RCRM [11] replaces

July 20, 2015 DRAFT


Frobenius norm with nuclear norm and considers solving the following optimization problem

minimizeVgk ,Ugk

G∑

g=1

‖[Ug1, . . . ,UgK ]HRg‖∗

subject to λmin(UHgkH(gk,g)Vgk) ≥ ǫ ∀g, k

(30)

where ǫ > 0. The constraint on the minimum eigenvalue of the received signal space ensures that: (a) all transmit and

receive beamformer matrices have rank d, and (b) the received signal space UHgkH(gk,g)Vgk is not rank deficient.

For any choice of ǫ > 0, the constraint on the smallest eigenvalue places a restriction on the received signal space

and the optimization is technically not a pure pursuit to align interference. While this is a necessary constraint

when channels are not generic, this constraint can be dropped for generic channels provided Ugk and Vgk are all

full rank. The optimization problem (30) is convex in either {Ugk} or {Vgk} when the other is held fixed and is

solved by alternately optimizing {Ugk} and {Vgk}. Each iteration requires solving two SDPs, one each to update

{Ugk} and {Vgk}.

In [16], instead of approximating rank using nuclear norm in the RCRM formulation, rank is approximated using

the log(det(·)) surrogate function. A majorization-minimization routine that minimizes a series of weighted nuclear

norms is proposed. The resulting algorithm consists of two loops— the outer loop updates the weights, while the

inner loop alternately optimizes transmit and receive beamformers for a fixed set of weights. The core optimization

problem solved at each step is similar to (30) and is given by

minimizeVgk,Ugk

G∑

g=1

‖Wg[Ug1, . . . ,UgK ]HRg‖∗

subject to λmin((UHgkH(gk,g)Vgk) ≥ ǫ ∀g, k.

(31)

where Wg is the weighting matrix. In the outer loop, the weighting matrices are updated as Pg(Σg + γI)−1PHg ,

where γ is a regularizing constant and PgΣgQHg is the SVD of the optimal [Ug1, . . . ,UgK ]HRg matrix obtained

from the inner loop. Note that unlike the reweighting rule in Section IV-B, the weight update rule in [16] does not

couple the weights. Further, since all the singular values of the interference matrices are to be reduced to zero, the

idea of coupling weights does not provide any benefit as the formulation then becomes equivalent to RCRM.

The ILM framework introduced in [8] serves as an important starting point for the subsequent algorithms discussed

above. The subsequent RCRM formulations are developed with the goal of inducing sparsity to reduce the dimensions

occupied by interference (after being projected by receive beamformers). When interference alignment is feasible,

the singular values of all the matrices in the objective of (30) and (31) are to be reduced to zero. However, this

places an extreme sparsity requirement on the optimization problems as the rank of a GKd×G(G− 1)Kd block

diagonal matrix needs to be reduced to zero. Typical rank minimization algorithms are not designed to generate

this level of sparsity and quite often do not succeed in completely eliminating interference.

As compared to ILM and RCRM, the new algorithms proposed in this paper have several key advantages.

First, by focusing on the received interference vectors before being projected by the receive beamformers, the

algorithms developed in this paper do not require alternately optimizing transmit and receive beamformers. Further,

July 20, 2015 DRAFT


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Inst

ance

so

fn

etw

ork

-wid

eal

ign

men

t

Number of iterations

RFNM SIR>60dBRFNM SIR>40dBRFNM SIR>20dB

AM SIR>60dBAM SIR>40dBAM SIR>20dB

ILM SIR>60dBILM SIR>40dBILM SIR>20dB

0 120 240 360 480 600

(a) Network-wide alignment

0

1

2

3

4

Nu

mb

ero

fit

erfe

ren

ce-f

ree

dim

ensi

on

s


RFNM SIR>60dBRFNM SIR>40dBRFNM SIR>20dB

AM SIR>60dBAM SIR>40dBAM SIR>20dB

ILM SIR>60dBILM SIR>40dBILM SIR>20dB

0 120 240 360 480 600

(b) Interference-free dimensions

Fig. 1: Alignment in the (4, 1, 3× 2) network using the RFNM, AM and ILM algorithms.

we incorporate prior knowledge of the requisite number of interference-free dimensions in designing our algorithms.

Such a design prevents the algorithms from converging to local minima and helps reduce the rank of the interference

matrices to the required extent.

VII. SIMULATION RESULTS

To evaluate the algorithms developed in this paper, we consider networks of varying sizes and number of antennas

as listed in Table I. The networks are chosen so that they lie on or below the proper-improper boundary d ≤

(M + N)/(GK + 1), and are known to be feasible. We assume d = 1 for each user in all cases, but different

networks have different total number of data streams and need different number of interference-free dimensions at

July 20, 2015 DRAFT


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Inst

ance

so

fn

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-wid

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ign

men

t


RFNM SIR>60dB

RFNM SIR>40dB

RFNM SIR>20dB

AM SIR>60dB

AM SIR>40dB

AM SIR>20dB

ILM SIR>60dB

ILM SIR>40dB

ILM SIR>20dB

0 12 24 36 48 60


0

1

2

3

4

Nu

mb

ero

fit

erfe

ren

ce-f

ree

dim

ensi

on

s


RFNM SIR>60dB

RFNM SIR>40dB

RFNM SIR>20dB

AM SIR>60dB

AM SIR>40dB

AM SIR>20dB

ILM SIR>60dB

ILM SIR>40dB

ILM SIR>20dB

0 12 24 36 48 60


Fig. 2: Alignment in the (4, 1, 3× 3) network using the RFNM, AM and ILM algorithms.

each BS. Note that for every network that is on the proper-improper boundary and has no redundant dimensions,

we also consider an identical network with redundant antennas on either the user or the BS side.

Since these algorithms are run on finite precision computers, a numerical threshold on interference suppression

is required to declare interference alignment. We adopt signal-to-interference (SIR) ratio as a metric to evaluate

interference suppression and declare interference to be aligned if SIR for every data stream in the network exceeds

a certain threshold. It is important to consider interference suppression across all data streams since there may exist

several local minima that eliminate interference only in a subset of the streams. While a high SIR threshold is an

accurate indicator of interference alignment, a lower threshold is of practical interest since most applications only

require interference to be suppressed down to the noise floor. We consider SIR thresholds of 20dB, 40dB and 60dB.

July 20, 2015 DRAFT


TABLE I

NETWORK PARAMETERS USED IN SIMULATIONS. IN ALL NETWORKS, THE NUMBER OF DATA STREAMS PER USER IS d = 1.

Network

(G,K,N ×M)

Redun-

dancy

Total no.

of data

streams

Interference-free

dimensions to be

created per BS

1(4, 1, 3× 2) No 4 1

(4, 1, 3× 3) Yes 4 1

2(3, 2, 4× 3) No 6 2

(3, 2, 4× 4) Yes 6 2

3(3, 3, 6× 4) No 9 3

(3, 3, 7× 4) Yes 9 2

We opt to show the number of interference-free dimensions and instances of network-wide alignment rather than

system-level performance metrics such as sum rate because the former is more directly related to the optimization

objective of this paper.

The algorithms are tested over at least 400 channel realizations with channel coefficients drawn from a complex

circularly symmetric Gaussian distribution with unit variance in each dimension. We compare the algorithms

developed in this paper with ILM proposed in [8], the RCRM formulation proposed in [11] and the reweighted

nuclear norm minimization approach proposed to solve RCRM (WRCRM) in [16]. The convex optimization

problems involved in these algorithms are solved using CVX, a package for specifying and solving convex programs

[38], [39]. All algorithms are run for a fixed number of iterations.

One update of the transmit and receive beamformers is considered to be one iteration of an algorithm. In the

case of ILM, each iteration requires computing G(K + 1) eigenvalue decompositions, while RCRM and WRCRM

require solving two SDPs per iteration. Each iteration of AM and RFNM requires solving one unconstrained

quadratic program while RNNM requires solving one SDP. Due to the significant differences in the per-iteration

complexities, the results corresponding to ILM, AM and RFNM algorithms are presented separately from the results

for RNNM, RCRM and WRCRM.

We first present results pertaining to the simplest two networks in Table I, the (4, 1, 3× 2) and the (4, 1, 3× 3)

networks.

A. The (4, 1, 3× 2) and (4, 1, 3× 3) Networks

The performance of the ILM, AM and RFNM algorithms on the (4, 1, 3× 2) network is plotted in Fig. 1. Note

that this network has no redundant antennas and requires one interference-free dimension at each BS. Fig. 1(a)

plots the number of instances (channel realizations) where network-wide alignment occurs as a function of the

number of iterations. Network-wide alignment requires the SIR in each of the four data streams to exceed a certain

July 20, 2015 DRAFT


0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Inst

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(4, 1, 3× 2) RNNM SIR>60dB

(4, 1, 3× 2) RNNM SIR>40dB

(4, 1, 3× 2) RNNM SIR>20dB

(4, 1, 3× 3) RNNM SIR>60dB

(4, 1, 3× 3) RNNM SIR>40dB

(4, 1, 3× 3) RNNM SIR>20dB

0 4 8 12 16 20 24 28 32 36 40


0

1

2

3

4

Nu

mb

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fit

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s


(4, 1, 3× 2) RNNM SIR>60dB

(4, 1, 3× 2) RNNM SIR>40dB

(4, 1, 3× 2) RNNM SIR>20dB

(4, 1, 3× 3) RNNM SIR>60dB

(4, 1, 3× 3) RNNM SIR>40dB

(4, 1, 3× 3) RNNM SIR>20dB

0 4 8 12 16 20 24 28 32 36 40


Fig. 3: Alignment in the (4, 1, 3× 2) and (4, 1, 3× 3) networks using the RNNM algorithm.

threshold. Fig. 1(b) plots the average number of interference-free data streams created as a function of the number

of iterations. In this case, we do not look for network-wide alignment and only focus on the SIR of each individual

data stream. Scenarios where partial interference alignment occurs (i.e., where interference is eliminated in only

some of the data streams) contribute towards interference-free dimensions (Fig. 1(b)) but not towards network-wide

alignment (Fig. 1(a)). It is seen that both AM and RFNM show faster convergence than ILM. For example, Fig. 1(a)

shows that after 120 iterations of each algorithm, RFNM achieves an SIR exceeding 60dB in all data streams for

65% of the channel realizations, but ILM only achieves an SIR exceeding 60dB in 40% of the channel realizations.

A similar trend is observed in Fig. 1(b). The performance difference between the different algorithms is minimal if

the algorithms are each run for more than 600 iterations. Thus, the main advantage of AM and RFNM over ILM

July 20, 2015 DRAFT


0

1

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3

4

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(4, 1, 3× 3) RNNM

(4, 1, 3× 3) RCRM

(4, 1, 3× 3) WRCRM

(4, 1, 3× 2) RNNM

(4, 1, 3× 2) RCRM

(4, 1, 3× 2) WRCRM

0 3 6 9 12 15 18 21 24 27 30

Fig. 4: Interference-free dimensions in the (4, 1, 3 × 2) and (4, 1, 3 × 3) networks using the WRCRM, RCRM and RNNM algorithms. The

SIR threshold is set to 20 dB.

is in convergence speed rather than ultimate performance.

The (4, 1, 3 × 3) network has a redundant antenna at each user and its impact is clearly seen in Fig. 2. ILM,

AM and RFNM all converge to aligned solutions within 50-60 iterations, with AM and RFNM again significantly

outperforming ILM. As compared to the (4, 1, 3× 2) network, adding a single additional antenna on the user-side

reduces the number of iterations required by almost a factor of 10, thus highlighting the significant benefit of having

redundant antennas when designing aligned beamformers.

Fig. 3 plots the performance of RNNM for the (4, 1, 3 × 3) and (4, 1, 3 × 2) networks. It is seen that aligned

beamformers are designed within 10-20 iterations in all cases. The performance of RNNM as compared to RCRM

and WRCRM is plotted in Fig. 4. In Fig. 4, we set the SIR threshold to 20dB and plot the number of interference-

free dimensions generated. In these two networks, a maximum of four such dimensions can be created. It is seen

that neither RCRM nor WRCRM generates the requisite number of interference-free dimensions; hence they not

achieve network-wide interference alignment. It appears that RCRM and WRCRM get trapped in local minima

within the first few iterations and are not able to reduce sufficient number of singular values to zero. This is true

even when the network has redundant antennas. This highlights the importance of designing algorithms that are

able to exploit the prior knowledge of the required rank.

B. The (3, 2, N ×M) and (3, 3, N ×M) Networks

To further test the observations made in the previous section we now consider networks where more interference-

free dimensions are required at each BS. Fig. 5 plots the performance of ILM, AM and RFNM on the second and

third set of networks listed in Table I that do not have redundant antennas. A broad trend that can be observed

from Fig. 5 is that as the network size increases, all three algorithms need several thousand iterations to design

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0%

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AM SIR>60dB

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AM SIR>20dB

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ILM SIR>40dB

ILM SIR>20dB

0 150 300 450 600 750 900 1050 1200 1350 1500

(a) (3, 2, 4× 3) network

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RFNM

0 240 480 720 960 1200 1440 1680 1920 2160 2400

(b) (3, 3, 6× 4) network

Fig. 5: Network-wide alignment using the RFNM, AM and ILM algorithms for two networks without redundant antennas.

aligned beamformers. Further, the performance gap between AM, RFNM and ILM grows smaller as the network

size increases.

Fig. 6 considers the same two networks but now appended with one redundant antenna at either the BS or the

user side. It is seen from Fig. 6 that the redundant antenna significantly reduces the the number of iterations required

to design aligned beamformers. In particular, AM and RFNM perform significantly better than ILM and require up

to 25% fewer iterations for a similar level of performance.

Fig. 7 plots the performance of RNNM for the same two networks with and without redundant antennas. It is

seen that unlike the previous three algorithms, RNNM is able to generate aligned beamformers, even for high SIR

thresholds within about 20-40 iterations, irrespective of whether there are redundant antennas or not. There is only

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AM SIR>40dB

AM SIR>20dB

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ILM SIR>20dB

0 10 20 30 40 50 60 70 80 90 100

(a) (3, 2, 4× 4) network

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AM SIR>60dB

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AM SIR>20dB

ILM SIR>60dB

ILM SIR>40dB

ILM SIR>20dB

0 30 60 90 120 150 180 210 240 270 300

(b) (3, 3, 7× 4) network

Fig. 6: Network-wide alignment using the RFNM, AM and ILM algorithms for two networks with redundant antennas.

a marginal increase in the number of iterations required as the network size grows. As before, we see in Fig. 8 that

the RCRM and WRCRM algorithms do not achieve the necessary number of interference-free dimensions, which

is six this case.

A conclusion drawn from these simulations is that networks with redundant antennas permit the use of algorithms

that are not necessarily well suited for sparsity but have the advantage of having low per-iteration complexity. When

this additional flexibility is absent, algorithms such as RNNM that are designed specifically to generate a desired

level of sparsity are necessary to design aligned beamformers. Hence, this trade-off between redundant antennas

and algorithmic complexity must be factored into design considerations for multi-antenna wireless networks.

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(3, 2, 4× 4) RNNM SIR>60dB

(3, 2, 4× 4) RNNM SIR>40dB

(3, 2, 4× 4) RNNM SIR>20dB

(3, 2, 4× 3) RNNM SIR>60dB

(3, 2, 4× 3) RNNM SIR>40dB

(3, 2, 4× 3) RNNM SIR>20dB

0 4 8 12 16 20 24 28 32 36 40

(a) (3, 2, 4× 3) and (3, 2, 4× 4) network

0%

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(3, 3, 7× 4) RNNM SIR>60dB

(3, 3, 7× 4) RNNM SIR>40dB

(3, 3, 7× 4) RNNM SIR>20dB

(3, 3, 6× 4) RNNM SIR>60dB

(3, 3, 6× 4) RNNM SIR>40dB

(3, 3, 6× 4) RNNM SIR>20dB

0 10 20 30 40 50

(b) (3, 3, 6× 4) and (3, 3, 7× 4) networks

Fig. 7: Network-wide alignment using RNNM.

VIII. CONCLUSION

In this paper we propose several new approaches to designing aligned beamformers in a cellular network based

on a reformulation of the conditions for interference alignment. Using these alternate conditions, we formulate a

rank minimization problem to design aligned transmit beamformers in the uplink and solve the rank minimization

using reweighted matrix norm minimization leading to RNNM and RFNM algorithms. A crucial aspect of these

algorithms is a novel reweighting rule that exploits prior knowledge of the required rank of interference matrices.

This paper also develops the AM algorithm for interference alignment where the prior knowledge of the rank of

the interference matrices is used to allow the rank-deficient matrices to be expressed as a product of two matrices,

followed by optimizing these matrices in an alternating manner to minimize a quadratic objective. Each iteration

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1

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(3, 2, 4× 4) RNNM

(3, 2, 4× 4) RCRM

(3, 2, 4× 4) WRCRM

(3, 2, 4× 3) RNNM

(3, 2, 4× 3) RCRM

(3, 2, 4× 3) WRCRM

0 3 6 9 12 15 18 21 24 27 30

Fig. 8: Interference-free dimensions in the (3, 2, 4 × 3) and (3, 2, 4 × 4) networks using the WRCRM, RCRM and RNNM algorithms. The

SIR threshold is set to 20 dB.

of this algorithm only requires solving an unconstrained convex quadratic program. In terms of complexity, while

RNNM requires solving an SDP in each iteration, RFNM and AM only require solving an unconstrained quadratic

program.

Simulation results indicate that RNNM is very effective in designing aligned beamformers irrespective of the

presence of redundant antennas. It significantly outperforms previously proposed RCRM and WRCRM algorithms

(with similar per-iteration complexity) in terms of maximizing the number of interference-free dimensions. In

contrast, RFNM and AM show significantly faster convergence in systems with redundant antennas when compared

to ILM (again a similar per-iteration complexity).

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95–110.

Gokul Sridharan (S’08-M’15) received his B.Tech and M.Tech (dual degree) in Electrical Engineering from the Indian

Institute of Technology Madras, Chennai, India, in 2008, and the M.A.Sc. and Ph.D. degrees in Electrical Engineering

from the University of Toronto, Toronto, Canada, in 2010 and 2014, respectively. His research interests include wireless

communications, convex optimization and information theory.

July 20, 2015 DRAFT


Wei Yu (S’97-M’02-SM’08-F14) received the B.A.Sc. degree in Computer Engineering and Mathematics from the

University of Waterloo, Waterloo, Ontario, Canada in 1997 and M.S. and Ph.D. degrees in Electrical Engineering from

Stanford University, Stanford, CA, in 1998 and 2002, respectively. Since 2002, he has been with the Electrical and

Computer Engineering Department at the University of Toronto, Toronto, Ontario, Canada, where he is now Professor

and holds a Canada Research Chair (Tier 1) in Information Theory and Wireless Communications. His main research

interests include information theory, optimization, wireless communications and broadband access networks.

Prof. Wei Yu currently serves on the IEEE Information Theory Society Board of Governors (2015-17). He is an

IEEE Communications Society Distinguished Lecturer (2015-16). He served as an Associate Editor for IEEE Transactions on Information

Theory (2010-2013), as an Editor for IEEE Transactions on Communications (2009-2011), as an Editor for IEEE Transactions on Wireless

Communications (2004-2007), and as a Guest Editor for a number of special issues for the IEEE Journal on Selected Areas in Communications

and the EURASIP Journal on Applied Signal Processing. He was a Technical Program co-chair of the IEEE Communication Theory Workshop

in 2014, and a Technical Program Committee co-chair of the Communication Theory Symposium at the IEEE International Conference on

Communications (ICC) in 2012. He was a member of the Signal Processing for Communications and Networking Technical Committee of the

IEEE Signal Processing Society (2008-2013). Prof. Wei Yu received a Steacie Memorial Fellowship in 2015, an IEEE Communications Society

Best Tutorial Paper Award in 2015, an IEEE ICC Best Paper Award in 2013, an IEEE Signal Processing Society Best Paper Award in 2008,

the McCharles Prize for Early Career Research Distinction in 2008, the Early Career Teaching Award from the Faculty of Applied Science and

Engineering, University of Toronto in 2007, and an Early Researcher Award from Ontario in 2006. Prof. Wei Yu was named a Highly Cited

Researcher by Thomson Reuters in 2014. He is a registered Professional Engineer in Ontario.

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